166 ANALYTICAL MECHANICS path. In this case the motion of the particle is not due to the force in question. 129. Work Done Against the Gravitational Force. — These special cases may be illustrated by considering the work done in raising a body from a lower to a higher level against the gravitational attraction of the earth. Consider the work done in taking a particle from A to B, along each of |he three paths shown in Fig. 93. (a) Suppose the particle to be taken from A to C and then to B] the work done in taking it from A to C comes wider Case IV. The direction of motion is at right angles to that of the gravitational force, therefore no work is done gfcgainst it. The work done in taking the particle from C to B comes under Case III; the force and the motion are in the same direction. Therefore the work done is inhere h is the vertical height of B above A. (b) Suppose the particle to be taken along the straight line AB. This comes under Case II. The angle between the force and the direction of motion is constant. Therefore W = mgl cos a, where Z is the length of the line AB. But since Z cos a = h, the work done is pIG. 93. the same as in (a), that is, mgh. (c) Suppose the particle to be taken along the curve AB. This comes under Case I. Then W = mg I cos a ds Jo = mg C JQ dh [since ds cos a = dh] = mgh.